# Basic definitions in Lie Theory.

This is the first of a series of posts on elementary Lie theory. My goal is to reach the classification of simple Lie algebras by Dynkin diagrams starting from scratch. I will first recall the various basic definitions.

**Definition****:** A *Lie algebra *is a vector space over some field together with a bilinear application

satisfying the two following axioms:

- for all
- for all

The second axiom is called the *Jacobi identity. *Since this application is not necessarily associative, it is essential to include the *Lie bracket *when writing the product. Note that bilinearity of the product and the first axiom give us

for all So multiplication in a Lie algebra is *anticommutative*. This means that for all

Here is an important

**Example: **Starting with any associative algebra over (i.e. a -vector space with an associative bilinear multiplication), we can create a Lie algebra by defining Then and

for all and By the same argument on the other variable, we get that is bilinear as hoped. Note that we only used bilinearity of multiplication in to show this. However, the associativity condition will be seen to be necessary (and sufficient) for the Jacobi identity to be valid. Indeed, for any we have

thus

Therefore, the Jacobi identity is satisfied if and only if multiplication in is associative.

**Definition: **For any algebra there is a linear map defined by

In fact, is nothing else than left multiplication by in

**Definition: **A *derivation *of an algebra is a linear map satisfying the Leibniz rule :

**Proposition:** Let be an algebra over satisfying axiom 1 ( for all with denoting multiplication in ). Then is a Lie algebra iff is a derivation of for all

*Proof*: Suppose in Then using the anticommutativity of the Jacobi identity reads

which is equivalent to

QED

We have showed that a Lie algebra is an anticommutative algebra in which left multiplication by any element is a derivation.

Let’s now state some more

**Definitions:** A *Lie algebra homomorphism *is a linear map between two Lie algebras such that and a

*Lie algebra isomorphism*is a bijective homomorphism. If are subsets of , define

So a *Lie subalgebra * of is a subspace such that , i.e. it is asked to be closed under multiplication in . An *ideal *of is a subspace such that .

Note that for two subspaces and , the anticommutativity of the bracket gives that (because for and , we have that so taking products of subspaces is commutative and there is no distinction between left and right ideals.

Finally, with a Lie algebra and an ideal of we can form the *quotient Lie algebra * by defining

**References: **

*Lectures on Lie Groups and Lie algebras*by R. Carter, G. Segal, I. MacDonald.*Lie Groups Beyond an Introduction*by W. Knapp.

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